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“International Portfolio Management” Ioannis Kalaitzis Supervisor: Nicolas Topaloglou Athens University of Economics and Business Department of International and European Economic Studies September 2020, Athens

“International Portfolio Management” · 2020. 10. 12. · “International Portfolio Management” Ioannis Kalaitzis Supervisor: Nicolas Topaloglou Athens University of Economics

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  • “International Portfolio Management”

    Ioannis Kalaitzis

    Supervisor: Nicolas Topaloglou

    Athens University of Economics and Business

    Department of International and European Economic

    Studies

    September 2020, Athens

  • 1

    Table of contents 1 Introduction ........................................................................................................................................... 3

    2 Kinds of risk .......................................................................................................................................... 4

    2.1 Market risk: ................................................................................................................................... 4

    2.2 Exchange risk: ............................................................................................................................... 4

    2.3 Interest rate risk: ........................................................................................................................... 4

    2.4 Credit risk: .................................................................................................................................... 4

    2.5 Liquidity risk: ................................................................................................................................ 5

    2.6 Inflation risk: ................................................................................................................................. 5

    2.7 Settlement risk: ............................................................................................................................. 5

    2.8 Sovereign risk: .............................................................................................................................. 5

    3 Risk measures and mathematical formulations ..................................................................................... 6

    3.1 Variance: ....................................................................................................................................... 8

    3.2 Value at Risk (VaR): ..................................................................................................................... 9

    3.3 Conditional Value at Risk (CVaR): ............................................................................................ 10

    3.4 Mean Absolute Deviation (MAD) .............................................................................................. 12

    3.5 Tracking error models ................................................................................................................. 13

    3.6 Put/Call Efficient Frontiers Model .............................................................................................. 14

    3.7 Expected Utility Maximization Model........................................................................................ 15

    4 International Diversification – Does it pay to go internationally? ...................................................... 18

    4.1 Empirical Results ........................................................................................................................ 19

    5 Empirical Application ......................................................................................................................... 26

    5.1 Data Description ......................................................................................................................... 27

    5.2 Methodology ............................................................................................................................... 27

    5.2.1 Domestic Portfolio .............................................................................................................. 27

    5.2.2 International Portfolio ......................................................................................................... 28

    5.3 Empirical results ......................................................................................................................... 29

    5.3.1 Efficient Frontier ................................................................................................................. 29

    5.3.2 Back Testing ....................................................................................................................... 31

    5.4 Analysis....................................................................................................................................... 34

    5.4.1 Descriptive statistics of returns ........................................................................................... 34

    5.4.2 Exchange rates .................................................................................................................... 35

    5.5 Home Asset Bias ......................................................................................................................... 35

    6 Conclusion .......................................................................................................................................... 37

    7 References ........................................................................................................................................... 38

  • 2

  • 3

    1 Introduction

    Risk is something infinitely dimensional and depends on all of the possible events in the world. Financial

    advisors, organizations and investors aim to structure their investment strategies combining assets that will

    offer a balance between risk and return. Since 1952, when Markowitz published his paper about portfolio

    selection in which he tries to maximize portfolio expected return for a given amount of risk, researchers

    have created various risk measures for optimal portfolio selection. However, it is not attainable to determine

    risk precisely. The only accurate answer we can give to the question “What is the maximal loss we can

    suffer over some time horizon?” is, “We can lose everything!” However, the probability of this event is

    very small.

    The current thesis is separated in a theoretical and an empirical part. As far as the theoretical part concerns,

    we analyze the various kinds of risks that an investor faces while investing in financial assets. Secondly,

    we cite the risk measures and their mathematical formulation that have been published in literature and are

    used commonly in modern economies for the portfolio selection. These measures are applied to financial,

    engineering and insurance sector by the managers in order to make optimal decisions. In the last section of

    the theoretical part, we investigate the benefits of investing internationally for investors who come from

    developed and development countries. The impact of international diversification depends on a variety of

    factors that we analyze in that section from a financial and an econometrical point of view, according to the

    international literature.

    As far as the empirical part concerns, we implement a portfolio optimization problem using CVaR model

    as risk measurement. Particularly, we apply the optimization problem on an investor from Germany who

    has the option to invest in a domestic and an international portfolio with assets from Germany, U.S.A and

    Japan. After receiving the historical data for the returns of the assets and the exchange rates for a selected

    period of time, we run the optimization problem on GAMS model, and we get the optimal weights for each

    asset. By determining the efficient frontiers and conducting a backing test for domestic and international

    portfolios, we reach to a conclusion about the profitability of the investments. In the end, we interpret the

    outputs, by analyzing the historical returns of the assets in tandem with the effect of exchange rates.

  • 4

    2 Kinds of risk The term “risk” describes the uncertainty that the realized return of an investment will not be equal to the

    expected return. The financial instruments contain various risks, according to their specific characteristics

    and the transactions among them comprise high risks of reducing or losing the initial (or multiple) investing

    capital. Thereby, the transaction with financial instruments is appropriate to investors who understand and

    interpret both the movements of these instruments and the undertaken risks. There are several forms of risk

    that affect the performance of a portfolio and the investors are trying to mitigate them.

    2.1 Market risk: is the risk which can cause fluctuations both to the fair value and to the future cash flows of the financial

    instrument. It is also common known as “Systematic risk” and is the probability of a loss as a result of

    negative fluctuations in the market. For instance, a decrease in the price of equities, interest rates,

    currencies, commodities, valuable metals and generally values that are traded in the market, can reduce the

    invested capital. Market risk can affect the whole market and the investors can mitigate it by investing in a

    variety of asset classes to delegate the possibilities.

    2.2 Exchange risk: is the risk of the impact on evaluating an investment in foreign currency, due to a fluctuation of exchange

    rates. Furthermore, we can define the Exchange risk as the risk that can be caused from the changes in

    exchange rates, affecting the fair value or the future cash flows of a financial instrument. Once an investor

    has to deal with assets that are exposed to foreign currencies, he is prone to unpredictable fluctuations of

    the currency’s price.

    2.3 Interest rate risk: contains the possibility the value of an investment to be undermined due to a change in interest rates. As

    interest rate rise equity’s and bond’s prices fall and vice versa. . The relationship between the interest rate

    and bond prices can be interpreted by opportunity risk. When an investor buys bonds, assumes that if the

    interest rate increases, he will miss out the opportunity of purchasing the bonds with better returns. In that

    case, the demand for the bonds with lower returns declines because of the new opportunities that appear.

    The choice of new bonds with higher return rates that are issued, are now more attractive for the investors,

    than the previous bonds that they hold. The interest rate risk is called “the source” of market risk as the

    fluctuations in interest rates affect the market in total. In conclusion, it is a risk through fair value and future

    cash flows can either be affected negatively or have intense fluctuations.

    2.4 Credit risk: is the risk of one of the two parts to abrogate its obligation, causing loss to the other part. It is known that

    a bond is a kind of loan, issued by a government or an organization. Although the risk of issuer’s default is

    the greatest factor of setting the appropriate borrowing rate, usually the market can predict it, especially in

    periods of growth. When an investor buys a bond or a kind of loan has expectations for a return. The return

    is a summary of interest rate risk and credit risk. The interest rate risk is reflected in reference rate and the

    credit risk in credit spread. The credit risk wasn’t able to be isolated until the appearance of credit options.

    Through these instruments, in a market, credit risk was attainable to be transferred to the investors who had

  • 5

    the willing to take it. (Through the composition of a bond and an asset swap, the credit risk could be

    isolated).

    The euro system has adapted rules and institutions for the assessment of credit risk among the countries.

    The rating agencies (Moody’s, Standard & Poor’s, Fitch e.t.c.) have the duty to assess credit risk of the

    countries and big companies, providing useful information to the investors. If a bond has a low rating (B or

    C), the issuer has a high risk of default. Conversely, if it has a high rating (AAA, AA, or A), it's considered

    to be a safe investment. Also, Central Banks like Deutsche Bundesbank, Banque de France, Banco de

    Espana, Oesterreichische National bank provide detailed reports for the credit assessment.

    2.5 Liquidity risk: stems from the lack of marketability of an investment that happens when a financial product cannot be sold,

    as a result of not being attractive to the investors. In that case, the investor might be unable to convert an

    asset into cash without giving up capital and income due to a lack of buyers or an inefficient market.

    Usually, it is reflected either in bid-ask-spreads (the amount by which the ask price exceeds the bid price

    for an asset in the market) or in big price movements. These facts depict uncertainty and insecure based on

    the fact that a borrower cannot meet its short-term debt obligations. The liquidity risk is used from the

    investors as a measurement of the level of risk within a company. In the level of financial institutions, their

    sustainability depends on the ability to meet their debt obligations without realizing great losses.

    2.6 Inflation risk: is the risk that inflation (the increase in the prices of a basket of selected goods and services over a period

    of time) will erode the value and expected profit of an investment. When we interpret the performance of

    an investment without taking into consideration the inflation, the outcome that turns out is the nominal

    return. However, an investor should take into account the real return of its investment which is linked to

    the purchase power. Bonds are the financial instruments that are most vulnerable to inflationary risk. Most

    of the times, bonds obtain a fixed coupon rate that does not increase. Thus, if an investor buys a bond that

    pays a fixed interest rate, but inflation rises more than the interest rate, then the investor faces losses. In

    case this situation will go on until the maturity of the bond, the holder will lose more purchasing power,

    regardless of how safe investment it is.

    2.7 Settlement risk: is the risk that one party will fail to deliver the terms of a contract with another party at the time of

    settlement. It can happen when one side fails to fulfill its obligations either by not delivering the underlying

    asset or by not paying the cash value of the contract. The organized markets try to confine this kind of risk

    by adopting the appropriate procedures. In not structured markets (out of money market) the investor faces

    the credit risk of the counterparty.

    2.8 Sovereign risk: contains the possibility a central bank will apply exchange rules that will affect negatively the worth of the

    FOREX (Foreign Exchange) contracts that have been done by foreign investors. A FOREX contract is a

    kind of currency transaction, where the two parties make an agreement in order to exchange two designated

  • 6

    currencies at a specific future time. It is a common way for the byer to protect from currency price’s

    fluctuations. Foreign exchange traders have to deal with the danger that a central bank will alter its monetary

    policy and as a result will affect currency trades. Another factor that causes sovereign risk is the political

    instability that takes place when a government refuses to adhere to the payments agreement of its sovereign

    debt.

    3 Risk measures and mathematical formulations

    Portfolio optimization problem is an important topic since the pioneering Markowitz work on optimal

    portfolio selection. Specifically, an investor is interested in determining the optimal combination of n risky

    assets in such a way that the obtained portfolio has minimal risk and maximal expected gain. Although the

    idea of risk seems to be intuitively clear, it is difficult to formalize it. There is an efficient way to measure

    and quantify risk in almost every single market. However, each method is deeply associated with its specific

    market and cannot be used directly to other markets. For instance, the most useful way to measure risk on

    an equity portfolio is the volatility of returns, whereas for a government bond is the interest rate risk.

    Furthermore, for portfolios of options, the risk can be measured by the rate of the change in price when

    only one of the parameters changes (like the underlying asset, volatility, time to maturity). These indices

    are suited to measure the very specific types of risk related to a specific financial instrument.

    Based on investors preferences and risk tolerance several alternative formulations are available that will be

    analyzed in the current thesis. Prior to presenting the various risk measures and the mathematical

    formulation of them, it is necessary to set the “scenario generation” which is the base of the analysis.

    Scenario generation:

    Scenarios are used in order to include and measure all the disparate sources of risk that affect the

    performance of portfolio. Each scenario depicts a range of different parameters and values that react on the

    portfolio. Thus, is a way to represent uncertainty that stems from the variations of the included parameters.

    Probabilities Scenarios 𝑟1 𝑟2 ….. 𝑟𝑛

    𝑃𝑟𝑜𝑏1 𝑆1 𝑟11 𝑟12 ….. 𝑟1𝑛

    𝑃𝑟𝑜𝑏2 𝑆2 𝑟21 𝑟22 ….. 𝑟2𝑛

    𝑃𝑟𝑜𝑏3 𝑆3 𝑟31 𝑟32 ….. 𝑟3𝑛

    …... ….. ….. ….. ….. …..

    𝑃𝑟𝑜𝑏𝑇 𝑆𝑇 𝑟𝑇1 𝑟𝑇2 ….. 𝑟𝑇𝑛

    Let us assume that we have an universe of n instruments (i = 1,2,…, n)

  • 7

    �̃� = (𝑟1̃, 𝑟2̃, 𝑟3̃, … , 𝑟�̃�)𝑇, where �̃� is the vector of the random instrument’s return, which are unknown

    at the time of portfolio selection and are treated as random variables.

    𝐸(�̃�) = �̅� = (𝑟1̅, 𝑟2̅, 𝑟3̅, … , 𝑟�̅�)𝑇 , where E(�̃�) is the Expected return or the mean of a random

    instrument.

    𝑥 = (𝑥1, 𝑥2, … , 𝑥𝑛)𝑇 , where x is the allocation of an investor’s budget in the assets.

    ∑ 𝑥𝑖 = 1𝑛𝑖=1 (budget constraint)

    𝑋 = {𝑥 ∶ 𝑥𝑇 1 = 1 , 𝑥 ≥ 0}, which is the basic portfolio constraints in vector notation by using the

    conformable vector 1 = (1,1, … ,1)𝑇of ones.

    𝑅(𝑥, �̃�) = 𝑥𝑇 ∙ �̃� = ∑ 𝑥𝑖 ∙ 𝑟�̃�𝑛𝑖=1 , where 𝑅(𝑥, �̃�) is the portfolio’s uncertain performance at the end

    of the holding period.

    𝛺 = {𝑠|𝑠 = 1,2, … , 𝑇} is the finite set of discrete scenarios.

    𝑃𝑟𝑜𝑏𝑠 > 0 is the probability under a particular scenario (s) take the returns �̃�

    𝐸[𝑅(𝑥, �̃�)] = 𝑥𝑇 ∙ �̅� = ∑ 𝑥𝑖 ∙ 𝑟�̅�𝑛𝑖=1 , where 𝐸[𝑅(𝑥, �̃�)] is the Expected performance of portfolio.

    𝐹(𝑥, 𝑛) = 𝑃𝑟𝑜𝑏(𝑅(𝑥, �̃�) ≤ 𝑛), where F(x,n) is the market portfolio.

    According to the above table:

    𝑅(𝑥, 𝑟1) = ∑ 𝑥𝑖 ∙ 𝑟𝑖1𝑛

    𝑖=1

    𝐸[𝑅(𝑥, 𝑟)] = ∑ 𝑃𝑟𝑜𝑏(𝑠) ∙ 𝑅(𝑥, 𝑟𝑠)𝑠𝑇𝑠=1

    We use the function 𝜑(𝑥, 𝑟) as a risk measure. The choice of the risk measure is related to the investor’s

    profile and his preferences. Then for a certain minimal expected portfolio return μ, the 𝜑-efficient portfolio is the result of the below problem solution:

    min 𝜑( 𝑥, 𝑟)

    s.t. 𝐸[𝑅(𝑥, 𝑟)] ≥ 𝜇

    Axiomatic investigation of risk measures was suggested by Artzner et al. [1]. Rockafellar [2]defined a

    function as a coherent risk measure in the extended sense if it meets the following axioms:

    1. Sub-additivity: 𝜑(�̃� + �̃�) ≤ 𝜑(�̃�) + 𝜑(�̃�)

    2. Homogeneity: 𝜑(𝜆 ∙ �̃�) = 𝜆𝜑(�̃�)

    3. Monotonicity: 𝜑(�̃�) ≤ 𝜑(�̃�) ⇒ �̃� ≪ �̃� (we prefer �̃� than �̃�)

    4. Risk-free condition:(�̃� − 𝑛 ∙ 𝑟𝑓) = 𝜑(�̃�) − 𝑛 , where 𝑟𝑓 is the riskless rate.

    A function is called a coherent risk measure if it satisfies axioms 1,2,3,4.

    It’s vital for the risk measure to satisfy the first axiom (sub-additivity) because it allows the

    diversification. Diversification is a risk management strategy that mixes a wide variety of

    investments within a portfolio. Thus, a diversified portfolio has a significant lower risk because of

    the limited exposure to any single asset or risk. Last but not least, sub-additivity and homogeneity

    ensure that the risk measure function is convex, which is consistent with the theory of risk aversion.

    After setting the scenario generation, the next section defines the risk measures and their basic properties

    through their mathematical formulation:

  • 8

    3.1 Variance: The Mean-Variance model, which was introduced by Markowitz in 1952 and he published the article

    Portfolio selection in Journal of Finance 1952 [3], laid the foundations of the contemporary risk

    management. The model presents two vital parameters with regard to portfolio optimization problems:

    Expect return and risk. Portfolios have to tackle the problem of how to allocate wealth among several assets.

    However, an investor always prefers to maximize the returns of the portfolio and concurrently to make the

    risk as small as possible. Nevertheless, a high return always accompanied with a higher risk. That problem

    is one of the important research fields in modern risk management. Markowitz applied variance (or standard

    deviation) as a measure of risk and the mathematical formulation of it is:

    𝜎2 = ∑ ∑ 𝑤𝑖𝑤𝑗𝜎𝑖𝑗

    𝑁

    𝑗=1

    𝛮

    𝑖=1

    In that case, in order to make the notation simpler we introduce weights, where 𝑤𝑖 is the proportion of the

    portfolio held in assets 𝑖 (𝑖 = 1, … , 𝑁) and 𝑤𝑗 is the proportion of the portfolio held in assets 𝑗 (𝑗 = 1, … , 𝑁).

    Also, 𝜎𝑖𝑗 is the covariance between assets between 𝑖 and 𝑗.

    The demise of the risk in a portfolio implies a low variance(𝜎2). Hence, the fundamental problem of a portfolio requires to minimize the variance with respect to a fixed expected return �̅�. This is known as the Markowitz model [4]

    min ∑ ∑ 𝑤𝑖𝑤𝑗𝜎𝑖𝑗

    𝑁

    𝑗=1

    𝛮

    𝑖=1

    s.t. ∑ 𝑤𝑖�̅�𝑖 = �̅�𝛮𝑖=1

    ∑ 𝑤𝑖 = 1

    𝑁

    𝑖=1

    Alternatively, we want to maximize the portfolio return given constraints and given the portfolio

    volatility:

    𝑚𝑎𝑥 ∑ 𝑤𝑖 �̅�𝑖

    𝑁

    𝑖,𝑗=1

    s.t. ∑ ∑ 𝑤𝑖𝑤𝑗𝜎𝑖𝑗 = 𝜎2𝑁

    𝑗=1𝑁𝑖=1

    ∑ 𝑤𝑖 = 1

    𝑁

    𝑖=1

    The volatility of the portfolio is a highly non-linear function of the weights, whereas most of the other

    constraints are linear. Also, the portfolio return is linear in the weights. The volatility of the portfolio is a

    highly non-linear function of the weights, while the majority of the rest constraints are linear, included the

    portfolio return in the weights. By considering the mathematics of optimization we are able to select the

    formulation of the problem. Thus, we prefer to minimize risk (the standard deviation or the variance of the

  • 9

    portfolio), instead of maximizing return given non-linear constraints (for the level of risk). We can explain

    that choice since it is easier to minimize a non-linear function with linear constraints than the alternative.

    Also, it is important to mention the one-to-one relation between the standard deviation and the variance of

    the assets. Because of that property we can minimize the variance and get the same results for the portfolio

    allocation as would get if we minimized the standard deviation. Furthermore, another reason we prefer to

    minimize the risk of the portfolio is that variance is a quadratic function and several convenient

    mathematical methods exist in order to optimize them. One of them is called the method of the Langrangean

    multipliers.

    After presenting the portfolio optimization problem, using the variance of a portfolio as a risk measure, it

    is important to mention the theoretical problems of Markowitz’s approach. Many researchers do not blindly

    depend on the Mean-Variance model of Markowitz as it is based on the assumptions that (for a risk averse

    investor):

    the distribution of the rate of return is multivariate normal.

    the utility function of is a quadratic function of the rate of return.

    However, none of these assumptions applies in practice, leading the researchers and investors to develop a

    plethora of models in order to tackle the Mean-Variance model’s defects. Thus, a lot of improved models

    from computational and theoretical view have been made. In the current thesis a plethora of these models

    are presented below.

    3.2 Value at Risk (VaR): Information about future return of a portfolio is unknown at the time of portfolio allocation, so every

    decision brings some risk. The Value at Risk is an important measure of the risk exposure of a given

    portfolio. It is now the main tool in financial industry and part of regulatory mechanism.

  • 10

    Let X be a random variable with the cumulative distribution function: 𝐹𝑋(𝑧) = 𝑃{𝑋 ≤ 𝑧}.X may have meaning of loss or gain, but in the particular situation X has meaning of loss.

    The VaR of X with confidence level α ∈[0, 1] is:

    𝑉𝑎𝑅(𝑥, 𝑎) = min{𝑢|𝐹(𝑥, 𝑧) ≥ 1 − 𝑎} = min{𝑢|𝑃[𝑅(𝑥, �̃�) ≤ 𝑢] ≥ 1 − 𝑎}, where 𝑎=95%(or 99%)

    The VaR of a portfolio is defined as the maximum loss that will occur over a given period at a given

    probability level. Since we are interested in the most unfavorable outcomes, we would like to measure the

    level of losses which will not be exceeded with some fixed probability. The critical level typically used for

    this fixed probability is either 5% or 1%.

    Historically, VaR was introduced by financial companies in the late 80’s. Since then this approach has

    become popular among practitioners, the academic community and – most significantly – among central

    banks.

    The greatest advantage of VaR is the recognition and the acceptance that has as a widely accepted standard.

    Also, it is a good basis for comparison and measurement of risk between portfolios and investments, since

    the industry and the regulators agreed to use it as the primary risk management measure. The two valuable

    factors of the success of the current model is its simplicity in use and the stability and the consistency of

    the results.

    However, VaR risk measure has a grave disadvantage since it does not control scenarios exceeding VaR.

    More specifically, VaR does not take into account the extreme tails of the distribution that contain large

    losses with very low probability (less than(1 − 𝑎) ∙ 100%. Thus, the indifference of VaR to extreme

    negative returns may be an undesirable property, allowing the investor to take extraordinary and undesirable

    risks with a very small chance. Although this indifference of VaR hides a risk for the investors, it may have

    a positive result in statistics as its estimators are robust and are not affected by excessive losses. Particularly,

    these huge losses may “confuse” the statistical estimation procedure and produce misleading results.

    Furthermore, VaR model has another important drawback that affects seriously the estimated risk.

    Specifically, the model does not consider the property of sub-additivity which is a vital condition for a risk

    measure to be coherent. The lack of sub-additivity depicts that VaR of two different investment portfolios

    may be greater than the sum of the individual VaRs. As a result, the property of “diversification” is not

    satisfied, and that fact limits the use of VaR.

    Last, VaR is relatively difficult to optimize due to be a non-convex discontinuous function and exhibits

    multiple local extrema. Although significant progress has been made in this direction, efficient algorithms

    for solving such functions are lacking.

    In conclusion, VaR is not a coherent measure of risk due to the facts below:

    It does not take into consideration the extreme tails of the distribution that entail big losses.

    It does not urge and, indeed, sometimes prohibit diversification.

    It is difficult to optimize it.

    In order to deal with these disadvantages and to avoid large errors in the final decision, an alternative risk

    measure was introduced.

    3.3 Conditional Value at Risk (CVaR): CVaR is a coherent risk measure that estimates the mean of the beyond-VaR tail region, which may be

    widely used in the optimization of the portfolio. VaR answers the question: what is the maximum loss with

  • 11

    the confidence level 𝛼 ∙ 100% over a given time horizon? Thus, its calculation reveals that the loss will exceed VaR with likelihood(1 − 𝛼) ∙ 100%, but no information is provided on the amount of the excess loss, which may be significantly greater. By contrast, CVaR is considered a more consistent measure of

    risk than VaR. CVaR supplements the information provided by VaR and calculates the quantity of the

    excess loss.

    Below, CVaR is defined as the conditional expectation of portfolio returns below VaR for continuous

    distributions, which measures the expected value of the (1 − 𝑎) ∙ 100% lowest portfolio’s returns:

    𝐶𝑉𝑎𝑅 = 𝐸[𝑅(𝑥, �̃�)|𝑅(𝑥, �̃�) ≤ 𝑉𝑎𝑅]

    However, the formula does not provide a coherent risk measure for discrete distributions as it gives a non-

    convex function in portfolio positions. Furthermore, another drawback in case there is a discrete distribution

    is the lack of a vital property that must be applied, sub-additive. In order to face these problems, we define

    CVaR model for general distributions with the formula:

    𝐶𝑉𝑎𝑅 = (1 −∑ 𝑃𝑟𝑜𝑏𝑠

    {𝑠 ∈ 𝛺|𝑅(𝑥, 𝑟𝑠) ≤ 𝑧}1−𝑎

    ) ∙ 𝑧 +1

    1−𝑎∙ ∑ 𝑃𝑟𝑜𝑏𝑠𝑅(𝑥, 𝑟𝑠){𝑠 ∈ 𝛺|𝑅(𝑥, 𝑟𝑠) ≤ 𝑧}

    , where 𝑧 =

    𝑉𝑎𝑅

    The priority of each investor is to minimize the losses of his investments. In order to achieve that, as VaR

    and CVaR tends more to the right part of the distribution than to the other direction where the bigger losses

    exist, the investment is more profitable and appealing to the investors. To solve this risk management

    problem:

    max 𝑧 −1

    1 − 𝑎∑ 𝑃𝑟𝑜𝑏𝑠 ∙ 𝑦𝑠

    +

    𝑠

    𝑖=1

    𝑠. 𝑡. 𝑅(𝑥, 𝑟𝑠) = ∑ 𝑥𝑖 ∙ 𝑟𝑖𝑠

    𝑛

    𝑖=1

    𝐸[𝑅(𝑥, 𝑟𝑠)] = ∑ 𝑃𝑟𝑜𝑏𝑠 ∙ 𝑅(𝑥, 𝑟𝑠)

    𝑆

    𝑠=1

    ≥ 𝜇

    𝑦𝑠+ ≥ 𝑧 − 𝑅(𝑥, 𝑟𝑠)

    𝑦𝑠+ ≥ 0

    ∑ 𝑥𝑖 = 1𝑛𝑖=1 , 𝑥𝑖 ≥ 0 ∀ 𝑖=(1, 2,…n)

    Where 𝑧 = 𝑉𝑎𝑅 and 𝑦𝑠+is an instrumental variable. Also, we define the expected portfolio return with 𝜇.

    At this point, it is crucial to define the instrumental variable which is a tool for the above optimization

    problem. Hence, for every scenario 𝑠 ∈ 𝛺:

    𝑦𝑠+ = max[0, 𝑧 − 𝑅(𝑥, 𝑟𝑠)]

    This variable turns 0 when the return of the portfolio exceeds VaR for a particular scenario. In contrast,

    when the portfolio return is below VaR the variable equals to the difference between VaR and the realized

    performance.

  • 12

    Through the above maximization problem, the “CVaR-efficient frontier” is generated, which is the result

    of the CVaR’s optimization. It applies on each expected return 𝜇, related to each discrete scenario. Concurrently, apart from the “CVaR-efficient frontier” the VaR is calculated, providing a completed view

    from the whole tail of the returns distribution which depicts all the chances, even the lowest ones, to realize

    losses. At the end, the investor is informed about the allocation of their budget and the structure of their

    portfolio since the optimized portfolio is produced.

    The use of CVaR, as risk measure, is common in a lot of industries due to its properties that turn it into a

    consistent and a coherent measure of risk. CVaR is a convex and continuous function, allowing the

    optimization to be done quite efficiently. Also, its popularity is based on its property to take into account

    and control scenarios exceeding VaR, on the extreme tails of the distribution. It allows the investors to

    protect from the undesired risk which can hide substantial losses.

    The problem of choice between the two risk measures (VaR and CVaR), especially in financial risk

    management, has been quite popular in academic literature. The differences in mathematical properties,

    simplicity of optimization procedures and the accuracy in calculation and control the risk are the main

    factors that affect an investor’s decision. VaR has achieved popularity as an appropriate risk measure in a

    plethora of engineering applications, including the financial sector. On the other hand, CVaR is heavily

    used in the insurance industry since the achievement of profits is related to the effective minimization of

    risk. However, there is no doubt that both are widely popular for risk measure so in the academic research

    as in the economy.

    3.4 Mean Absolute Deviation (MAD) The MAD model is a risk measure which is widely used to solve large-scale portfolio optimization

    problems. It was introduced by Konno and Yamazaki (1991) due to the standard Mean-Variance model

    (MV) of Markowitz couldn’t deal with huge models. The innovative model uses the absolute-deviation of

    the rate of return of a portfolio instead of the variance to measure the risk. Although these two measures

    are almost the same from mathematical view, they have significant differences from computational view.

    Specifically, MAD model can be optimized by a linear program, whereas MV leads to a convex quadratic

    programming problem. The former can be solved distinctively faster than the latter and this is one of the

    main reasons we prefer MAD model.

    The definition of MAD model describes risk as the mean absolute deviation of portfolio return from its

    expected value. As it is conspicuous from the mathematical formulation:

    𝑀𝐴𝐷(𝑥) = 𝐸{ |𝐸[𝑅(𝑥, �̃�)] − 𝑅(𝑥, �̅�)|}

    When we refer to a discrete scenario set instead of a random asset’s return:

    𝑀𝐴𝐷(𝑥) = ∑ 𝑃𝑟𝑜𝑏𝑠 ∙ |𝐸(𝑅) − 𝑅𝑠|

    𝑆

    𝑠=1

    We could define the MAD model as the average of the positive distances of each point from the mean. The

    larger the MAD, the greater variability there is in the data. In order to optimize MAD, we need to solve the

    following linear problem:

    min ∑ 𝑃𝑟𝑜𝑏𝑠 ∙ 𝑦𝑠

    𝑆

    𝑠=1

    s.t. 𝑅(𝑥, 𝑟𝑠) = ∑ 𝑥𝑠 ∙ 𝑟𝑠𝑁𝑠=1

  • 13

    𝐸[𝑅(𝑥, 𝑟𝑠)] = ∑ 𝑃𝑟𝑜𝑏𝑠 ∙ 𝑅(𝑥, 𝑟𝑠)

    𝑁

    𝑠=1

    ≥ 𝜇

    ∑ 𝑥𝑖 = 1𝑛𝑖=1 , 𝑥𝑖 ≥ 0∀ i=(1, 2,…,n)

    𝑦𝑠 ≥ 𝐸[𝑅(𝑥, 𝑟)] − 𝑅(𝑥, 𝑟𝑠) ∀ 𝑠 = 1,2, … , 𝑆

    𝑦𝑠 ≥ 𝑅(𝑥, 𝑟𝑠) − 𝐸[𝑅(𝑥, 𝑟)] ∀ 𝑠 = 1,2, … , 𝑆

    𝑦𝑠 ≥ 0

    Comparable to CVaR model, we introduce an instrumental variable to linearize the absolute value

    expression. By setting the above conditions, we ensure that 𝑦𝑠 gathers the positive difference of the mean,

    either a bit is higher or lower than this. In case 𝑦𝑠 < 0 then it equals 0. The notion we follow is the same as it was in CVaR model. By solving the linear problem, the MAD- efficient frontier is generated according

    to the different values of the expected return 𝜇. For each expected return 𝜇 the investor manages the risk, since the output of MAD model is the allocation of an investor’s budget. Thereby, we have achieved to

    solve large- scale portfolio optimization problems with a wider computational simplicity than using the MV

    model and to build the efficient portfolio for each 𝜇, based on the MAD-efficient frontier. These models have been used in a plethora of portfolio optimization problems because of the groundbreaking method that

    was introduced in the 90’s.

    3.5 Tracking error models In the tracking error models, we introduce a different concept for allocation of the capital in comparison to

    the previous ones. In this case, the professional money manager (institutional investor) has to structure the

    portfolio according to a pre-specified benchmark. In other terms, the goal of a tracking error portfolio model

    is to replicate, as close as possible, a given benchmark. Therefore, the portfolio returns must relate to the

    performance of the imitated target. The allocation decision problem is based on the difference between the

    manager's return and the benchmark return, the so-called tracking error. The tracking-error optimization

    problems could be solved by finding a portfolio with maximum expected performance relative to the

    benchmark for a given tracking error volatility. For the implementation, it is necessary to control the

    deviations between the portfolio returns and the target. To achieve it, we impose an infinite penalty for any

    deviations that are more than a user specified parameter 𝜖𝑅(𝑥, �̅�) below the target. The condition contains

    both the cases where deviations being within 𝜖𝑅(𝑥, �̅�) below or above the benchmark and greater than it. Thus, we do not take into consideration returns that do not comply with that condition. The next step is to

    set the mathematical formulation:

    max ∑ 𝑝𝑠𝑅(𝑥, 𝑟𝑠)

    𝑆

    𝑠=1

    s.t. 𝑥 ∈ 𝑋

    𝑅(𝑥, 𝑟𝑠) ≥ 𝑔𝑠 − 𝜖𝑅(𝑥, �̅�), 𝑠 = 1,2, … , 𝑆

    where 𝑔𝑠is the return of the benchmark for each scenario 𝑠 ∈ 𝛺.

    It is conspicuous that the condition affects the downside risk which is defined with respect to the target.

    The target may be set equal to the return of an alternative investment opportunity or to a fixed value and

    thereby the target is the desired return we want the portfolio to achieve. Parameter 𝜖 is determined by the user and it is utilized to estimate the weights of each asset. In other words, the parameter is used to produce

  • 14

    the optimal portfolio. For that reason, it is essential to set an appropriate 𝜖 that is suitable with the data of

    the problem. The goal is to find the parameter with which we will produce the optimal portfolio for each

    scenario. In order to structure a portfolio close to the target, it is conducive to set the parameter equal to a

    small number because as we can observe from the above condition a low 𝜖leads to a portfolio return closer

    to the selected return we want to achieve. In conclusion, the parameter𝜖 depicts that the smaller it gets, the closer our portfolio approaches the benchmark.

    3.6 Put/Call Efficient Frontiers Model The main notion of the model is to build up a portfolio that trades off the risk (downside) against the reward

    (upside) given a benchmark (target). It is known that the portfolio’s reward is calculated through the

    expected return of the portfolio. The main goal of that model is to create a portfolio that performs better

    than the target for each scenario. In order to achieve that, it is crucial to measure both the portfolio’s

    outperformance from the target and the underperformance from the target. The tools that will provide all

    the needed information are two auxiliary variables counting the positive and the negative deviations of the

    portfolio return from the benchmark. Thus, we introduce 𝑦+ in order to measure the portfolio returns and

    𝑦− in order to captivate the risk:

    𝑦+ = max[ 𝑅(𝑥, �̃�) − �̃�, 0]

    𝑦− = max[ �̃� − 𝑅(𝑥, �̃�), 0]

    These auxiliary variables have a crucial role as they distinguish the model from the other risk measures. As

    it is conspicuous from the name of the model, the payoffs of the upside potential on the future portfolio

    return are similar to a call option, having as a strike price the price of the target. In contrast, the payoffs of

    the downside payoffs on the future portfolio return are similar to those of a short position in a put option.

    In other terms, when the return of the portfolio is above the target, the gains are identical to the payoffs of

    an exercised call option. The investor gains the difference between the performance of the portfolio and the

    price of the benchmark. Obviously in that case, there is a zero- downside potential and the put option is not

    exercised. On the contrary, when the portfolio return is below the target, the downside payoffs are identical

    to those of a short position in a put option. Then, the investor gains the difference between the price of the

    benchmark and the low portfolio performance. Again, in that case, the call option cannot be exercised as

    the upside potential is zero. The following figure depicts the Put/Call Efficient Frontier Model:

    𝑅(𝑥, 𝑟𝑠)𝑅(𝑥, 𝑟𝑠)

    𝑔𝑠

    Also, 𝑦+ is used to measure the upside potential of the portfolio to outperform the benchmark, while 𝑦−is used as a measurement of the downside risk of the portfolio. In discrete scenario setting we can express the

    definitions of the auxiliary variables as systems of inequalities:

    𝑦+𝑠 ≥ 𝑅(𝑥, 𝑟𝑠) − 𝑔𝑠, 𝑦+

    𝑠 ≥ 0, ∀𝑠 ∈ 𝛺

    𝑦+ 𝑦−

  • 15

    𝑦−𝑠 ≥ 𝑔𝑠 − 𝑅(𝑥, 𝑟𝑠), 𝑦−

    𝑠 ≥ 0, ∀𝑠 ∈ 𝛺

    The goal of the model is to produce portfolios that achieve the higher call for a given put. Under this

    occasion, the investment is considered to have the greatest performance and the portfolio is called put/call

    efficient. By definition, a portfolio is efficient when it offers the highest expected return for a defined level

    of risk or the lowest risk for a given level of expected return. The mathematical formulation of the Put/Call

    Efficient Frontier Model in discrete scenario setting is represented with a linear program for tracing the

    efficient frontier.

    max ∑ 𝑝𝑠𝑦+𝑠

    𝑆

    𝑠=1

    s.t ∑ 𝑃𝑟𝑜𝑏𝑠𝑦−𝑠 ≥ 𝑤𝑆𝑠=1 , w: risk target

    𝑦+𝑠 ≥ 0 ∀𝑠

    𝑦−𝑠 ≥ 0 ∀𝑠

    𝑦+𝑠 − 𝑦−

    𝑠 − [𝑅(𝑥, 𝑟𝑠)] − 𝑔𝑠] = 0, ∀𝑠 ∈ 𝛺

    The output of the maximization of the model is the portfolio of which the returns are above the target,

    limiting at the same time the downside losses.

    3.7 Expected Utility Maximization Model

    The previous models optimized a pre-specified measure of risk against a pre-specified measure of reward,

    such as the expected return of the portfolio. However, the investor’s choices and behavior have a vital role

    in portfolio selection and in setting a target return. In order to take into account such a need in portfolio

    optimization, we use the expected utility maximization model which allows the users to optimize according

    to their own criteria when trading of risk and rewards. The current model bases on the Expected Utility

    criterion which takes into consideration the risk preferences of an investor. It derives from the ability of a

    concave curve to depict the preferences of a risk-averse investor. According to that criterion, we can classify

    two mutually excluded investment plans because of their expected utility, since the criterion choose the one

    with the highest expected utility.

    Furthermore, the expected utility criterion can classify investment plans that have different risk, in spite of

    having the same expected return. Although the same level of expected returns implies same utility for the

    risk-averse investor, it would be wrong if we were indifferent between these two choices. Expected utility

    gives a solution on that selection problem, as it considers the risk preferences of the investor.

  • 16

    The above figure illustrates two investment plans having the same expected returns:

    𝐸(𝑌 + 𝑍�̃�) =1

    2(𝑌 − 𝑍𝐴) +

    1

    2(𝑌 + 𝑍𝐴) = 𝐸(𝑌 + 𝑍�̃�) =

    1

    2(𝑌 − 𝑍𝐵) +

    1

    2(𝑌 + 𝑍𝐵) = 𝑌

    but also, different variance as plan B shows higher deviations than A does. However, both of them depict

    the same utility for the expected returns:

    𝑈[𝐸(𝑌 + 𝑍�̃�)] = 𝑈[𝐸(𝑌 + 𝑍�̃�)] = 𝑈(𝑌)

    The conclusion is that the investor can distinguish which of the two investment plans appeals to his profile

    only by using the expected utility criterion.

    From a mathematical view, the ability of the criterion to classify the investment plans with different risks

    and the same returns is attributed to the concavity of the utility function. Consequently, the expected utility

    has different prices for two plans that have the same expected performance. For a concave utility function,

    the higher expected utility of the plan with the lower deviation is consistent to the investing behavior of a

    risk-averse investor. Such a behavior is caused because of that plan includes a risk of lower loss in

    comparison to the alternative plan, since the returns of the latter have higher variance.

    From the view of a portfolio, the expected utility criterion can be generalized for each scenario s, where

    𝑠 ∈ 𝛺. Thus, the expected utility is defined as the expected average price of the utility levels for each

    scenario:

    𝑉[𝑆1, 𝑆2, … , 𝑆𝑁; 𝑃𝑟𝑜𝑏1, 𝑃𝑟𝑜𝑏2, … , 𝑃𝑟𝑜𝑏𝑁] = ∑ 𝑃𝑟𝑜𝑏𝑠 ∙ 𝑈[𝑅(𝑥, 𝑟𝑠)]

    𝑁

    𝑠=1

    While the models of the previous sections are specialized to produce frontiers of efficient portfolios –from

    which the user has to select one- the expected utility maximization model allows the user to choose a unique

    portfolio that maximizes the user’s utility function. The maximization problem is presented as follows:

    𝑈(𝑌 − 𝑍𝐵)

    𝑌 − 𝑍𝐵

    𝑈(𝑌 + 𝑍𝐵)

    𝑌 + 𝑍𝐵

    𝑈(𝑌 + 𝑍𝐴)

    𝑌 + 𝑍𝐴

    𝑈(𝑌 − 𝑍𝐴)

    𝑌 − 𝑍𝐴

    𝑈(𝑌)

    𝑌

    𝐸[𝑈(𝑌 + 𝑍�̃�)] =1

    2⁄ 𝑈(𝑌 − 𝑍𝐵) +1

    2⁄ 𝑈(𝑌+𝑍𝐵)

    𝐸[𝑈(𝑌 + 𝑍�̃�)] =1

    2⁄ 𝑈(𝑌 − 𝑍𝐴) +1

    2⁄ 𝑈(𝑌+𝑍𝐴)

    𝑈(𝑌)

    𝑌

  • 17

    max ∑ 𝑃𝑟𝑜𝑏𝑠 ∙ 𝑈[𝑅(𝑥, 𝑟𝑠)]

    𝑆

    𝑠=1

    s.t 𝑥 ∈ 𝑋

    𝑅(𝑥, 𝑟𝑠) = ∑ 𝑥𝑖𝑟𝑖𝑠

    𝑛

    𝑖=1

    ∑ 𝑥𝑖 = 1

    𝑛

    𝑖=1

  • 18

    4 International Diversification – Does it pay to go internationally?

    In this part, we utilize the available literature and models to determine the benefits from investing overseas.

    Specifically, we investigate how the benefits of international portfolio diversification differ across the

    countries, taking into consideration a plethora of factors that could affect the result. Also, the analysis

    includes the short selling constraints in investments that are imposed in many developing countries to

    control capital flows. The outputs appear to be statistically significant and provide strong evidence about

    the advantages of international diversification.

    The first research we are going to analyze comes from Joost Driessen and Luc Laeven [5] who carried out

    an investigation about how the benefits of international portfolio diversification differ across countries from

    the perspective of a local investor. Particularly, they investigate for a large cross-section of countries with

    both large and small stock markets, what benefits occur by adding international stock investment

    opportunities for a domestic investor that invests in local stocks only. They also measure the economic size

    of these benefits. Although the literature on international portfolio diversification takes a U.S. perspective,

    this paper takes the viewpoint of domestic investors in many different countries, since the benefits of

    investing abroad for investors in countries with less well-diversified and developed stock markets as those

    in the United States are likely to benefit more from investing abroad than U.S investors do. Also, the writers

    consider both the case of frictionless markets, where the investors don’t face any costs to short assets, and

    the case where short selling constraints prevail especially in developing countries. Furthermore, the results

    are calculated both in local currencies since we consider a local investor that uses it, and in U.S. dollar as

    in practice, investors in many counties use it as their base currency.

    The analysis bases on two types of diversification opportunities appeared for an investor that currently

    invests in the local equity index:

    A. Regional diversification: A regional equity index for the country’s region

    B. Global diversification: MSCI indices for the US, Europe and Far East.

    The reason for distinguishing between these two scenarios is that it is interesting to see how much of the

    international diversification benefits can be achieved by investing regionally and globally.

    To measure the benefits of diversification for domestic investors, we use Markowitz’s notion about the

    standard mean-variance, assuming either a mean-variance utility function for the investor, or normally

    distributed asset returns. The concept is to examine if an addition in portfolio’s assets will shift the mean-

    variance frontier.

    In order to measure the economic size of diversification benefits, we use the increase in Sharpe ratios and

    in expected return that is obtained when adding assets to the investment set. An alternative way to measure

    it is through the expected return. If the expected return is increased when adding to the portfolio the new

    𝑁 assets given the same (or lower) variance as for the optimal portfolio of 𝐾 benchmark assets, then the

    diversification benefits are economic significant.

    To calculate the statistical significance of the diversification possibilities, they use the regression tests for

    mean-variance spanning developed by Huberman, Gur and Kandel (1987) [6] , DeRoon, Nijman, and

    Werker (2001) [7], and DeRoon, Frans and Nijman (2001) [8]. In that test, it is examined whether adding

    𝑁 new assets to a given set of K benchmark assets leads to a significant shift in the mean-variance frontier.

    In the research, the domestic index is the K benchmark assets, so that K equals to 1, and the 𝑁 additional

  • 19

    assets are given by asset sets A or B (either regional or global diversification is implemented). In case of

    frictionless markets, the test can be performed using the regression:

    𝑟𝑡+1 = 𝑎 + 𝛽𝑅𝑡+1 + 𝜀𝑡+1

    Where 𝑟𝑡+1 is a 𝑁-dimensional column vector with the 𝑁 returns on the additional assets, 𝑅𝑡+1 is a 𝐾-

    dimensional return vector for the 𝐾 benchmark assets, 𝑎 is a 𝑁-dimensional constant term, 𝛽is a 𝑁 by 𝐾

    matrix with coefficients and 𝜀𝑡+1is a 𝑁-dimensional vector with zero-expectation error terms.

    The null hypothesis that the 𝐾 benchmark assets span the entire market of all 𝐾 + 𝑁 assets is equivalent to the restrictions:

    𝑎 = 0 𝑎𝑛𝑑 𝛽𝜄𝛫 = 𝜄𝛮

    where 𝜄𝛮 is a 𝑁-dimensional vector with ones.

    If the equations hold, the return on each asset that we add to the portfolio can be decomposed into the return

    on a portfolio of benchmark assets and a zero-expectation error term that is uncorrelated with the benchmark

    portfolio return. Hereby, from the view of mean-variance framework, such an additional asset can affect

    the variance of the portfolio return and not the expected return and the investor will not include the

    additional asset in his strategy. In other terms, if the spanning hypothesis holds, the optimal mean-variance

    portfolio consists of the 𝐾 benchmark assets.

    Also, it is vital to analyze how global diversification benefits change over the time. They use the country

    risk measure to interpret the correlations of local returns with global indices and the variances of local

    indices.

    Finally, as far as the data concerns, for developed countries the writers use the stock market index developed

    by Morgan Stanley Capital International (MSCI) and for developing countries they use the S&P/IFC Global

    Index from Standard & Poor’, on a monthly base for a period 1985 to 2002. The research includes 23

    developed markets and 29 developing markets. Also, they download country risk ratings for each country

    s reported by the International Country Risk Guide maintained by the Political Risk Services group and

    they use the ICRG composite index as a general measure of country risk. The rating ranges from 0 to 100,

    with a higher rating implying less country risk. Furthermore, they download monthly currency data from

    Datastream in order to returns in U.S.D. terms and in terms of the local currency.

    4.1 Empirical Results First of all, they estimate both increases in Sharpe ratios and in expected returns given variance, when a

    regional index or three global MSCI indices are added to a local stock index portfolio, in order to measure

    the diversification benefits for the local investors of the 7 regions (Latin America, Asia, Eastern Europe,

    Middle East & Africa, Europe 16, Pacific, North America. The results for each region are divided into two

    types of diversification opportunities: Regional Diversification and Global Diversification. Also, the Sharpe

    ratios are calculated for each country both with and without short-selling constraints and in terms of local

    currency and U.S.D. returns. Short-selling constraints are either applied to all countries or only to

    developing countries. The following table reports the results summarized by averaging country-by-country

    results within each region.

  • 20

    Table 1: Increase in Sharpe ratio and Expected return by region

    It is conspicuous from the Table 1 that the regional diversification benefits appear largest on average for

    countries in Eastern Europe, even when allowing for short selling constraints. For instance, the expected

    return for Eastern European countries of investing in the region is 0.291% per month (3.492% per year)

    higher than the return of the investment in the local market only, in U.S.D. terms, even when short selling

    constraints prevail.

    For the majority of the investors of the seven regions, the global diversification benefits are significantly

    higher than the regional diversification benefits, according to both of Sharpe ratio and Expected returns,

    with the largest benefits appear for the countries in Eastern Europe. These figures suggest that the economic

    size of global diversification benefits is large for most countries, since the increases in Sharpe ratio and in

    expected returns implies economic significance of the diversification benefits.

    In a plethora of developing countries, short-selling constraints are imposed due to the government restricts

    the investors to invest in their home country, which is potentially costly for these investors, since these

    restrictions reduce the diversification possibilities. Therefore, the writers use short-selling constraints in

    order to have more realistic results. However, it is obvious from the table that the results of global

    diversification benefits are not affected significantly when assuming short sales constraints in developing

    countries, due to for almost all countries it is optimal to invest a part of the total asset portfolio to the local

    stock market index. But when we assume short sails constraints for all countries, which is a less realistic

    assumption than the imposed constraints in developing countries, there is a substantial decrease in the

    benefits of global diversification for the majority of the countries, especially when diversification benefits

    are measured in terms of increasing in expected returns. Nevertheless, the global diversification benefits

    for most of the regions are substantial and statistically significant as they were without short sales

    constraints.

  • 21

    Finally, as far as the currency concerns, there isn’t a significant difference in the results of diversification

    benefits when we interpret them either in U.S.D. terms or in the local currency terms.

    The next step is to calculate the statistical significance of the results by using the regression tests for mean-

    variance spanning. Table 2 depicts the results of the spanning tests on a regional basis. Specifically, it

    includes the percentages of cases in which the spanning hypothesis is rejected.

    Table 2: Rejection of Spanning Tests by Region

    When the global diversification takes place, the spanning hypothesis is rejected for all countries at 1%

    confidence level, both when the short sales constraints are imposed to the investors in developing countries

    and when the investors do not face any constraints at all. That means the global diversification benefits are

    statistically significant for domestic investors around the world, due to tiny p-values that force us to reject

    the spanning hypothesis for all confidence levels.

    However, when we introduce short-selling constraints for all countries, which is not a realistic scenario, the

    spanning hypothesis is not rejected for several countries in the regions, proving that the constraints

    disappear both the regional and global diversification benefits for a plethora of countries. Despite that fact,

    there are still many countries that gain benefits from diversification.

    In the next section, the writers aim to interpret which factors affect the benefits of international

    diversification in the various countries. Particularly, they explore whether the country risk and other country

    characteristics have an effect on the gains from international diversification. In order to test it, they use a

    cross-country regression as a mean to recognize the factor that affects the benefits. They use as dependent

    variable the difference between the Sharpe ratio of the global portfolio (including the local index) and the

    Sharpe ratio of the local index so as to measure the economic benefits for a mean-variance investor who

    invests not only in the local but also in the global market. The higher figure this variable indicates the larger

    benefits the investor gains. The regression is the following:

    𝐼𝑆𝑅𝑖,𝑡 = 𝑎 + 𝑏𝐼𝐶𝑅𝐺𝑖,𝑡 + 𝑐𝑆𝑀𝐶𝑖,𝑡 + 𝑑𝑇𝑟𝐺𝐷𝑃𝑖,𝑦 + 𝑒𝑃𝐶𝐺𝐷𝑃𝑖,𝑡 + 𝜀𝑖,𝑡

    Where 𝐼𝑆𝑅 is the Increase in Sharpe Ratio, 𝑎 is a constant, 𝐼𝐶𝑅𝐺 is the country risk rating, 𝑆𝑀𝐶 is the

    Stock Market Capitalization in U.S.D, 𝑇𝑟𝐺𝐷𝑃 is the trade to GDP, 𝑃𝐶𝐺𝐷𝑃 is the private credit to GDP

  • 22

    and 𝜀 is the error. The dependent variables are calculated using stock market return and interest rate data

    for the period 1985-2002.Generally, these variables proxy for the level of institutional development in the

    country. It is expected that more development countries benefit less from international portfolio

    diversification. The results of the regression are presented in Table 3:

    Table 3: Cross-country differences in the benefits of foreign diversification

    In comparison to all dependent variables used in the estimation, they found only the Country risk rating to

    be statistically significant for the increase in the Sharpe ratio variable. Also, a vital output is that the

    international diversification benefits are larger for countries with higher country risk. Nothing special

    changes when we interpret the results in local currency terms than in U.S.D. terms and the short selling

    constraints do not affect the results. To control robustness, they excluded the region of East Asia and

    Russian due to the sample period covers both the Asian and Russian debt crisis of 1997 and the results did

    not change significantly.

    Except for the paper of Driessen and Laeven, another research carried out by Wan-Jiun Paul Chiou [10]

    analyzes the benefits of international diversification by using two different measures in order to estimate

    them. Specifically, the writer utilizes two straightforward measures: the increase in risk-adjusted premium

    by investing in the maximum risk-adjusted return (MRR) portfolio and the reduction in the volatility by

    investing the minimum variance portfolio (MVP) on the international efficient frontiers. The mathematical

    formulation of the increment in risk-adjusted premium requires setting the maximum risk-adjusted return

    (MRR). Thus, if there is no investment restriction neither for the developing nor the developed markets:

    𝑀𝑅𝑅 = max {(𝑤𝑝

    𝑇𝜇

    𝑤𝑝𝑇𝑉𝑤𝑝

    )

    12⁄

    |𝑤𝑝𝑇 ∈ 𝑆}

    Where 𝑆 is the set of all real vectors 𝑤 that define the weights such that 𝑤𝑇1 = 1. The mean and variance-covariance of asset returns can be expressed as a vector 𝜇 and a positive definite matrix 𝑉, respectively.

    From the above maximization problem, we expect to take the weights of the various assets in the portfolio.

    However, the investor takes into consideration various constraints before investing in a foreign market and

    the mathematical formulation can be obtained by the following:

  • 23

    𝑀𝑅𝑅 = max {(𝑤𝑝

    𝑇𝜇

    𝑤𝑝𝑇𝑉𝑤𝑝

    )

    12⁄

    |𝑤𝑝𝑇 ∈ 𝑃𝐽}

    Where 𝑃𝐽 = {𝑤𝑝 ∈ 𝑆 ∶ 0 ≤ 𝑤𝑖 ≤ 𝐽𝑤(𝐶𝑎𝑝)𝑖 ≤ 1, 𝑖 = 1,2, … , 𝑁}, 𝐽 is any number greater than one and

    𝑤(𝐶𝑎𝑝)𝑖 is the weight of the market value of each country 𝑖. By setting the MRR, the greatest increment of unit-risk performance brought by international diversification, for the domestic investor, is:

    𝛿𝑖 = 1 −𝑅𝑅𝑖

    𝑀𝑅𝑅

    Where 𝑅𝑅𝑖 is the risk-adjusted performance of domestic portfolio in country 𝑖. The second measure used for the estimation of the international diversification benefits is the reduction in volatility. Elton and Gruber

    (1995) support that the estimation of the expected returns may hide a difficulty for the investors and for

    that reason they seek to minimize the variance of a portfolio. A common way to minimize the variance of

    a portfolio is by obtaining the vector of weights of the minimum-variance portfolio (MVP), 𝑤𝑀𝑉𝑃. According to the research of Li et al. (2003) [12], the maximum decline in volatility of domestic portfolio

    in country 𝑖, by international diversification is:

    𝜀𝑖 = 1 − [𝑤𝑀𝑉𝑃

    𝛵 𝑉𝑤𝑀𝑉𝑃𝑉𝑖

    ]

    2

    Where 𝑉𝑖is the variance of return of the domestic portfolio. The data used in this study are the U.S. Dollar-denominated monthly returns of the MSCI indices for 21 developed countries and 13 developing countries

    from January 1988 to December 2004. The differences of the benefits of global diversification for the

    developed and developing countries with various investment constraints are reported below:

  • 24

    Table 4: Portfolio weights and benefit from global diversification

    The significant cross-market differences in risk-adjusted premiums, implies that domestic investors in the

    countries of low performance can improve the mean-variance efficiency of their portfolios by optimally

    allocating funds to overseas assets. According to Table 4, the countries where the investors can gain the

    highest mean-variance efficiency by investing internationally are Japan, Philippines, Thailand, New

    Zealand, Portugal, Indonesia and Korea. Also, local investors can eliminate their portfolio’s volatility by

    investing in strategies of MVP. The general conclusion is that investments from developing markets have

    substantial margins of benefits by investing internationally, in contrast to investors that come from

    developed countries.

    According to the MRRP index, 6 countries only are proposed to invest in, whereas the MVP suggests 11

    countries. Furthermore, the weights in small capital markets are excessive, while in the major financial

    markets are significantly low. Based on these two indices, an investor should invest 80% of fund in

    Argentina, Chile, Denmark, Mexico and Switzerland, which represent only 4% of global market value and

    20% in the US markets. The results comply with the outcomes of Driessen and Laeven, since both of them

    depict that there are substantial benefits for the investors of developing countries by investing globally.

    Finally, the writer sites in Table 5 the unconstrained efficient frontiers composed by global portfolios,

    countries of different developmental stages and various regions.

  • 25

    Table 5: Efficient Frontiers without constraints

    The different positions of international portfolios composed by various groups of countries depict different

    impacts on mean-variance efficiency. Specifically, the efficient frontiers that derive from developed

    countries, Europe and North America are of low risk and low return, while the efficient frontiers of

    investments in emerging markets and Latin America are of high risk with high return.

    In conclusion, the two researches provide useful information about the differed benefits of international

    portfolio diversification across countries from the perspective of a local investor. The first finding is that

    there are substantial benefits from investing both regional and global for domestic investors in both

    developed and developing countries, even under the realistic assumption that investors cannot short sell

    stocks in developing markets. However, the benefits of international portfolio diversification are larger for

    developing countries than for developed ones. Concerning the factors that play a crucial role in

    determination of the benefits, country risk is vital, since countries with higher country risk have a greater

    potential benefit of global diversification. The results are statistically significant. Another output of the

    investigation is the decreasing trend of diversification benefits across the countries, due to the

    improvements in country risk over time.

  • 26

    5 Empirical Application

    In this section we implement a portfolio optimization problem of an investor who invests firstly in domestic

    market and secondly in international capital markets. The problem is applied on an investor based in

    Germany which means that the first part of the empirical application refers to the German market only,

    while in the second one two foreign markets are added. We picked the United States of America (U.S.A)

    and Japan as the two countries where the investor can invest in, except for his country, because of the

    different level of development of the their financial markets. The assets which the investor can select for

    his portfolio are 1 stock index and 2 bond indices from each country. Thus, in the case of the domestic

    portfolio optimization problem, the portfolio can be consisted of up to 3 assets, whereas the international

    portfolio up to 9 assets.

    In order to calculate and evaluate the performance of both domestic and international portfolio during the

    selected past period, we implement a method called back testing. Back testing is a methodology for

    assessing the viability of a strategy by discovering how it would perform using historical data. Particularly,

    it displays the level in which a portfolio would have performed ex-post and as a result it provides strong

    evidence about the outcome of the investment when implemented in reality. The notion is that a portfolio

    that would have performed poorly in the past will probably perform poorly in the future, and vice versa.

    Thus, in our sample we implement back testing in order to assess the performance of the optimal portfolio

    during a long period of time and to come to a conclusion about the future results of that specific investment.

    However, before implementing the back testing, it is necessary to calculate the optimal weights of each

    asset that an investor should invest in. In order to achieve it, we run a CVaR model on GAMS system which

    solves the portfolio optimization problem.1 Hence, by holding the optimal weights of the assets we can

    export the real returns of the portfolio that were performed in the past.

    Apart from the back testing we construct the efficient frontier for the domestic and the international

    portfolio since it is a way to understand how a portfolio’s expected returns vary considering the amount of

    risk taken. Specifically, the efficient frontier is the locus between the returns and the CVaR (risk) of all the

    combinations of the portfolios that offer the highest expected return for a defined level of risk. Those

    portfolios that lie below the efficient frontier are sub-optimal because they do not provide enough return

    for risk assumed. The efficient frontier is a commonly known path for the investors to select investments

    since it provides information for all the kinds of investors. According to efficient frontier, a risk-seeing

    investor would select portfolios that lie on the right end of the efficient frontier, since there are located

    portfolios that are expected to have a high degree of risk coupled with high potential returns. Contrarily,

    risk-averse investors will select portfolios that lie on the left end of the efficient frontier.

    Last, it is vital to interpret the factors that affect the asset’s optimal weighs coming from the CVaR’s model

    solution, so as to understand the results from a financial point of view. The two main reasons in international

    portfolio which contribute to the optimal level of each asset are the descriptive statistics of the indices’

    returns and the exchange risk since the investor based in Germany has to deal with assets that are exposed

    to U.S. Dollar and Yen. Apparently, in the case of domestic portfolio the exchange risk does not exist, due

    to the fact that the investments share joint currency.

    1 The GAMS (General Algebraic Modeling System) is a high-level modeling system for mathematical programming. It can formulate complex and large-scale optimization models in a notation similar to their algebraic notation. Gams is designed to solve linear, non-linear and mixed-integer optimization problems and thus it can use many different solvers.

  • 27

    5.1 Data Description As referred, we are considering 3 assets from each country (1 stock index and 2 bond indices). Particularly,

    as far as Germany concerns, we focus on the DAX 30 index, on FTSE German Government Bond Index 1-

    3 year sector and on FTSE German Government Bond Index 7-10 year sector. Furthermore, regarding to

    U.S.’s financial markets, we examine S&P 500 index, the US FTSE World Government Bond Index

    (WGBI) US, 1-3Y and the US FTSE WGBI US, 7-10Y. Last, in Japan, we take into consideration the

    NIKKEI 225 Stock index, the FTSE Japanese Government Bond Index 1-3 year sector and the FTSE

    Japanese Government Bond Index 7-10 year sector. We have collected historical prices for all the indices

    starting from 31/12/2004 until 30/4/2020, on a monthly basis. Thus, we were able to compute the returns

    for each asset through all the months of our period. Also, due to the investments in foreign currencies, we

    have collected data regarding to exchange rates, specifically the exchange rate for U.S. Dollar (U.S.D.) to

    Euro and the rate for Yen to Euro. Again, the exchange rates are available on a monthly basis, the same

    periods as the 9 assets were.

    5.2 Methodology In this part is made an analytical description of the steps that were followed using the GAMS model. First

    of all, we selected the CVaR mathematical model in order to solve the optimization problem. CVaR is a

    very popular coherent risk measure for optimal portfolio selection, because it goes beyond the information

    revealed by VaR and thus it takes into account the distribution of the tail, where the losses can be huge

    there.

    5.2.1 Domestic Portfolio Efficient Frontier: As referred previously, we need to find the locus between the returns and the

    standard deviation (risk) of all the combinations of the portfolios that offer the highest expected return for

    a defined level of risk. The data which we import in the GAMS model are returns of the 3 German assets

    for the last 10 years (120 observations). In the first step, we find the bottom end of the efficient frontier’s

    curve, where the risk is at its lowest level as it relates to return. In order to calculate that point in GAMS

    model, we maximize CVaR model. Thus, we get the expected return of the portfolio which lies on the left

    end of the efficient frontier. In the second step, we export the upper limit of the efficient frontier which lies

    on the right tail, by solving the problem of maximization of total returns on GAMS model. After these steps,

    we hold the two extreme points on the efficient frontier’s curve which display the minimum and the

    maximum expected returns. In order to determine the rest points, we separate the gap between the extreme

    points in 8 equal spaces, getting 8 different expected returns. Then, we run the CVaR model 8 times, by

    assuming every time one of the 8 expected returns that we calculated previously, as the expected portfolio’s

    target return. Therefore, we locate the middle points and the efficient frontier is completed.

    Back testing: In order to implement this dynamic test, for each specific month we run the model,

    we have to adjust the historic data to the optimization problem. The period of implementation starts from

    April 2017 until April 2020, which means that we solve the optimization problem 37 times (37 months).

    For every month we run the model, we insert a fixed number of historical data which is 147, since we got

    data for the returns of the assets for 147 months before April 2017 (the beginning of the test). Thus, the

    sample of historical data is determined as the last 147 observations prior to the month we examine. For

    example, when solving the optimization problem for June 2018, the inserted returns in the model are from

    March 2006 until May 2018. After solving the portfolio optimization problem through GAMS model, we

    get the optimal weights for the 3 assets that an investor should invest in, monthly. In order to evaluate the

  • 28

    performance of the investment for all the months of the test, we multiply the optimal weights with the

    realized returns in the specific month we examine. We repeat that procedure for all the tested months and

    in the end, we have created the monthly gains or losses of the portfolio and by summarizing them we get

    the final performance of the portfolio. Therefore, we are able to evaluate the return of a portfolio consisted

    of 3 German assets the last 3 years, utilizing the CVaR model as a mean of selecting the optimal portfolio.

    5.2.2 International Portfolio Efficient frontier: We implement the same methodology as we did in domestic portfolio, including

    however the exchange rate of the foreign currencies. In order an investor from Germany to invest in

    American and Japanese capital markets, it is necessary to convert euro in the currency that each asset can

    be traded in. Thus, we need to find a mechanism for changing one currency into another, to be able to invest

    in foreign assets. Our aim is to detect the return of each asset for all the months in the sample. The notion

    is the following:

    In order an investor from Germany to invest 1€ in a foreign country (USD or Yen), he has to convert it in

    the specific currency. If the exchange rate is USD to Euro (or Yen), the type to convert 1€ in USD (or in

    Yen) is:

    1 ∗ 𝑒𝑐

    𝑒𝑐: is the exchange rate in country c

    Also, the performance of the asset 𝑖 in scenario 𝑛 in 1 month is:

    1 ∗ 𝑒𝑐 ∗ (1 + 𝑅𝑖,𝑐𝑛 )

    where 𝑅𝑖,𝑐𝑛 is the return of asset 𝑖 in scenario n.

    In order to convert the performance in € (domestic currency):

    1 ∗ 𝑒𝑐 ∗ (1 + 𝑅𝑖,𝑐𝑛 )

    𝑒𝑐𝑛

    where 𝑒𝑐𝑛 is the exchange rate in scenario 𝑛.

    In conclusion, the performance of the asset 𝑖 is:

    𝑒𝑐 ∗ (1 + 𝑅𝑖,𝑐𝑛 )

    𝑒𝑐𝑛 − 1

    However, this type is applied on our test since the exchange rates are USD to Euro and Yen to Euro. In case

    the rates were expressed vice versa (Euro to Yen and Euro to USD), the performance of the asset 𝑖 is:

    𝑒𝑐𝑛 ∗ (1 + 𝑅𝑖,𝑐

    𝑛 )

    𝑒𝑐− 1

    By utilizing this type, we are able to create the returns for all the foreign assets on a monthly basis. Then,

    we follow the same procedure as we did in domestic portfolio. The only different point is that the portfolio

    is consisted of 9 assets, instead of 3 as it was in domestic portfolio. Hence, through the steps we

    implemented in domestic portfolio related to maximization of CVaR model in GAMS, we draw the efficient

    frontier of the international portfolio.

  • 29

    Back Testing: We repeat the methodology as we did in domestic portfolio by adding the 6 foreign

    assets. In order to take into account the exchange rates that affect the returns of the investment in foreign

    assets, we have to implement the steps that we did previously, so as to export the returns gained by an

    investor from Germany. Hence, the notion of the back testing remains the same and we examine the

    performance of the international portfolio, according to the returns performed in the past.

    5.3 Empirical results We present below the results of the efficient frontier and back testing for the domestic and the international

    portfolio.

    5.3.1 Efficient Frontier

    Table 6: Domestic Efficient Frontier

    Figure 1: Efficient Frontiers

    0

    0,005

    0,01

    0,015

    0,02

    0,025

    -0,14 -0,12 -0,1 -0,08 -0,06 -0,04 -0,02 0

    International EfficientFrontier

    DomesticEfficientFrontier

    Efficient Frontiers

    CVa

    Expected Returns

    Domestic Efficient Frontier

    CVaR Expected Returns

    Bottom end -0,00814 0,01897

    m=0,001105 -0,00815 0,01899

    m=0,00214 -0,00816 0,01901

    m=0,003175 -0,00819 0,01903

    m=0,00421 -0,00822 0,01905

    m=0,005245 -0,00824 0,01907

    m=0,00628 -0,00827 0,01909

    m=0,007315 -0,0083 0,01911

    Top end -0,03 0,01913

    International Efficient Frontier

    CVaR Expected Returns

    Bottom end -0,00425 0,00007

    m=0,001105 -0,00599 0,00111

    m=0,00214 -0,00922 0,00214

    m=0,003175 -0,01317 0,00318

    m=0,00421 -0,01725 0,00421

    m=0,005245 -0,02532 0,00525

    m=0,00628 -0,04367 0,00628

    m=0,007315 -0,0648 0,00732

    Top end -0,125 0,00835

    Table 7: International Efficient Frontier

  • 30

    In Table 6 and Table 7 are the results that we received from the GAMS model, after solving the optimization

    problem using CVaR as risk measure. These results are represented in Figure 1 and by connecting all the

    points we get the efficient frontier for each portfolio.

    It is obvious from the graph that Domestic portfolio does not have significant volatility in expected returns,

    since in the bottom end of the efficient frontier it is 0,01897 while in the top end it is 0,01913. The risk is

    computed -0,00814 and -0,03 respectively, which shows that the German investor does not face a

    substantial risk by investing in domestic portfolio. Furthermore, the efficient frontier of the international

    portfolio displays that the maximum return an investor from Germany can gain is 0,00835 whereas the

    minimum return is 0,00007. Also, the minimum price of CVaR (risk) is -0.125 and the maximum is -

    0.00425.

    Comparing the expected returns and the CVaR of the portfolios, we can observe that there is a positive

    correlation between expected returns and CVaR. Specifically, in domestic portfolio there is higher CVaR

    than in international one, due to the higher expected returns. The interpretation is attributed to the definition

    of CVaR. Since CVaR quantifies the expected losses that occur beyond the VaR breakpoint, it is more

    likely to be higher in a portfolio with larger expected returns. On the contrary, in international portfolio the

    potential returns are low for a German investor and therefore they are associated with low levels of extreme

    losses.

  • 31

    5.3.2 Back Testing

    Domestic Portfolio

    Portfolio’s

    returns

    Cumulative

    portfolio’s returns

    Optimal

    Weight S1

    Optimal

    Weight S2

    Optimal

    Weight S3

    Months -0,000293672 1 0,01342 0,98658 0

    April ‘17 -0,000381804 0,999706328 0,01342 0,98658 0

    May ‘17 -0,003693683 0,999324635 0,01342 0,98658 0

    June ‘17 0,00096082 0,995633447 0,01342 0,98658 0

    July ‘17 0,000610809 0,996590071 0,01342 0,98658 0

    Aug. ‘17 -0,00022372 0,997198797 0,01342 0,98658 0

    Sept. ‘17 0,000868697 0,996975704 0,01342 0,98658 0

    Oct. ‘17 -0,001730693 0,997841774 0,01342 0,98658 0

    Nov. ‘17 -0,001634322 0,996114816 0,01342 0,98658 0

    Dec. ‘17 -0,002161012 0,994486843 0,01342 0,98658 0

    Jan. ‘18 -3,90672E-05 0,992337745 0,01342 0,98658 0

    Feb. ‘18 -0,000111754 0,992298977 0,01342 0,98658 0

    Mar. ‘18 -0,000309253 0,992188084 0,01342 0,98658 0

    April ‘18 0,001752117 0,991881247 0,01342 0,98658 0

    May ‘18 -0,001120883 0,993619139 0,01342 0,98658 0

    June ‘18 -0,001354472 0,992505408 0,01342 0,98658 0

    July ‘18 -0,000283257 0,991161087 0,01342 0,98658 0

    Aug. ‘18 -0,00176035 0,990880334 0,01342 0,98658 0

    Sept. ‘18 0,000521172 0,989136038 0,01342 0,98658 0

    Oct. ‘18 -0,000827312 0,989651548 0,01342 0,98658 0

    Nov. ‘18 -0,001137356 0,988832798 0,01449 0,98551 0

    Dec. ‘18 -0,000723619 0,987708143 0,01449 0,98551 0

    Jan. ‘19 -0,00065075 0,986993419 0,01449 0,98551 0

    Feb. ‘19 0,00104477 0,986351133 0,01449 0,98551 0

    Mar. ‘19 0,000114 0,987381643 0,01449 0,98551 0

    April ‘19 0,000441616 0,987494204 0,01449 0,98551 0

    May ‘19 0,001436525 0,987930297 0,01449 0,98551 0

    June ‘19 -0,000212691 0,989349484 0,01449 0,98551 0

    July ‘19 0,001543428 0,989139059 0,01449 0,98551 0

    Aug. ‘19 -0,00304921 0,990665724 0,01449 0,98551 0

    Sept. ‘19 -0,001696144 0,987644976 0,01449 0,98551 0

    Oct. ‘19 -0,000892558 0,985969788 0,01449 0,98551 0

    Nov. ‘19 -0,001240872 0,985089753 0,01449 0,98551 0

    Dec. ‘19 0,000417119 0,983867382 0,01449 0,98551 0

    Jan. ‘20 -7,67644E-05 0,984277772 0,01449 0,98551 0

    Feb. ‘20 -0,004440918 0,984202214 0,01449 0,98551 0

    Mar. ‘20 0,001023008 0,979831453 0,01351 0,98649 0

    April ‘20 -0,000293672 0,980833828 0,01342 0,98658 0 Table 8: Domestic portfolio - back testing

  • 32

    International Portfolio Months Portfolio

    ’s returns

    Cumulative

    portfolio’s

    returns

    Opt. W.

    S1

    Opt. W.

    S2

    Opt.

    W.

    S3

    Opt.

    W.

    S4

    Opt.

    W.

    S5

    Opt.

    W.

    S6

    Opt. W.

    S7

    Opt.

    W.

    S8

    Opt.

    W.

    S9

    Apr. ‘17 -0,00029 0,99970633 0,01342 0,98658 0 0 0 0 0 0 0

    May ‘17 -0,00038 0,99932464 0,01342 0,98658 0 0 0 0 0 0 0

    June ‘17 -0,00369 0,99563345 0,01342 0,98658 0 0 0 0 0 0 0

    July ‘17 0,00096 0,99659378 0,01305 0,9868 0 0 0 0 0,00015 0 0

    Aug. ‘17 0,00064 0,99723358 0,01128 0,98769 0 0 0 0 0,00103 0 0

    Sep. ‘17 -0,00002 0,99721284 0,00799 0,98569 0 0 0 0 0,00632 0 0

    Oct. ‘17 0,00083 0,99803731 0,01127 0,9877 0 0 0 0 0,00102 0 0

    Nov. ‘17 -0,00172 0,99632185 0,00794 0,98578 0 0 0 0 0,00628 0 0

    Dec. ‘17 -0,00153 0,99480088 0,01021 0,98185 0 0 0 0 0,00794 0 0

    Jan. ‘18 -0,0022